1. The problem statement, all variables and given/known data Suppose we can calculate a quantity f(t) and we need its Fourier transform F(w). Looks to me that accuracywise Filon's rule, e.g. approximating the computed f(t) by splines and analytically integrating piecewise should be more accurate than an FFT, at least for a smooth f(t); especially for large enough w(>w_Nyquist) the FFT would show aliasing, while Filon's rule might lose accuracy due to the summation of a large number of terms of possibly different sign. Why is FFT always used?